On generalized renewal measures and certain first passage times.(English)Zbl 0759.60088

The author considers generalized renewal functions $$U_ \varphi(t)=\sum_{n\geq 0}\varphi(n)P(S_ n\leq t)$$ for arbitrary random walks $$S_ n$$ with positive drift $$\mu$$ and functions $$\varphi$$ for which $$t^{-\alpha}\varphi(t)$$ is slowly varying. The main result states that for $$\alpha\geq 0$$, $$E((X^ -_ 1)^ 2\varphi(X_ 1))<\infty$$ is equivalent to the finiteness of $$U_ \varphi(t)$$ (for all or for some $$t)$$, and also to $$(t\varphi(t))^{-1}U_ \varphi(t)\sim(\alpha+1)^{- 1}\mu^{-\alpha-1}$$, as $$t\to\infty$$; if $$\alpha=0$$, it must be assumed additionally that $$\varphi$$ is ultimately increasing. The above moment condition still implies the other statements if $$\alpha\in(-1,0]$$ and $$\varphi$$ is ultimately decreasing. Similar results are given for the increments $$U_ \varphi(t+h)-U_ \varphi(t)$$ and the sequence of maxima $$M_ n=\max_{0\leq j\leq n}S_ j$$ instead of $$S_ n$$.

MSC:

 60K05 Renewal theory 60G40 Stopping times; optimal stopping problems; gambling theory
Full Text: